Archive for April 2025

Revering Musings on String Compactification (but mostly de Sitter)

Posted on Tuesday, April 22, 2025

With Vaibhav Kalvakota, I wrote some notes on string compactification starting from supergravity compactification and ending with whether there exists a de Sitter solution in string theory in non-classical cases. This review was meant to initiate discussions on the de Sitter case.


Abstract. These notes are written on the (realistic) string compactifications and the string de Sitter vacua problem. A lot of unanswered questions remain in these regime which are highlighted using a historical canvas and exposition. We discuss also the KKLT proposal and other recent discussions around if there is a de Sitter vacua? In the exposition, we review the Calabi-Yau manifolds, supergravity compactification, flux compactification, moduli stabilization, and all that. This has been written in the same series where we wrote on the de Sitter quantum gravity and observables, ‘Revering Musings on de Sitter and Holography, 2023’.

The preprint is available at this link now.

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The Quill 15 ~ Properties of Morphisms

Posted on Tuesday, April 1, 2025

This will be a very quick act.
Grothendieck emphasized that properties of objects are better understood than the properties of morphism. But his relative perspective also lighted a switch of perspective. Vakil notes that it is a change of perspective while algebraic geometry is still intact. For the properties of a scheme, like quasi-compact, separated, affine, and so on, we can translate them to the properties of morphisms. For the affine case, we can use the Affine Communication Lemma, so let us take that example here.


Let us say that P is a property of a scheme (so it is going to be an object here) of being affine (that it is isomorphic to Spec A). Then the a morphism of schemes
\pi: R \to S
can be said to have this property if \pi is affine. How do we know that? We can say that for every subscheme U \in S, the object \pi^{-1} (U) should have the property P (in this case, being affine). Using the Affine Communication Lemma, we can check it for every subset and verify the properties locally on the base scheme S.
Now, while it is similar to the saying that every fiber on S should have property P, it is more than this fiber-by-fiber analysis. It behaves well when fiber moves. Such a fiber is more of a scheme-theoretic fiber. Such a change of perspective should be helpful, as I understand, in the moduli spaces (and deformation theory) discussion where there is a morphism of universal family, and its properties should determine the properties of the moduli space. A property will ensure that these fibers behave well when you move on the base moduli space.

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